K-MEANS ALGORITHM VIA PREPROCESSING TECHNIQUE AND SINGULAR VALUE DECOMPOSITION FOR HIGH DIMENSION DATASETS
DAUDA USMAN
A thesis submitted in fulfilment of the requirements for the award of the degree of
Doctor of Philosophy (Mathematics)
Faculty of Science UniversitiTeknologi Malaysia
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ACKNOWLEDGEMENT
I would like to express my greatest appreciation to my advisor, Assoc. Prof. Dr. Ismail Bin Mohamad, for introducing me to the world of clustering and provided me with a steady stream of his insights into data visualization and manipulation. I feel blessed to have him as my advisor. He helped me not only accomplish my dream of becoming a professional statistician but also develop a more mature personality. I thank him for every piece of his intensive efforts that have been put into this research work.
I would also like to thank the management team of Umaru Musa Yar’adua University, Katsina- Nigeria for giving me the opportunity to further my study.
I owe many thanks to my collaborators for this and other material: Aishah Bint Mohd Noor Universiti Malaysia Perlis, Malaysia and Abdulrahman from Syria. There is much that never would have been discovered without their insight and their diligence. I would also like to thank my colleagues at UTM; it was their presence that made UTM the great research environment it was.
ABSTRACT
vi
ABSTRAK
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEME iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES xi
LIST OF FIGURES xiii
LIST OF ABBREVIATIONS xv
LIST OF APPENDICES xvi
1 INTRODUCTION 1
1.1 Background of the Study 1
1.2 Problem Statement 3
1.3 Objectives of the Study 4
1.4 Scope of the Study 5
1.5 Contribution of the Study 6
1.6 Thesis Organization 6
1.7 Chapter Summary 7
2 LITERATURE REVIEW 8
2.1 Introduction 8
2.2 Cluster Analysis 9
2.3 The Clustering Process 9
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2.4.1 Euclidean Distance 13
2.4.2 Manhattan Distance 13
2.4.3 14
2.4.4 Mahalanobis Distance 14
2.5 K-means Cluster Analysis 15
2.6 Principal Component Analysis (PCA) 17
2.7 Cluster Initialization Technique 18
2.8 Data Standardization 20
2.9 Detecting Outlier and Cleaning Data 21
2.10 Cluster Validity 22
2.11 Fundamental Concepts of Cluster Validity 23
2.12 Some Related Work on K-means 24
2.13 Rerearch Framework 29
2.14 Chapter Summary 31
3 METHODOLOGY 32
3.1 Introduction 32
3.2 Pre-processing Method 32
3.2.1 Data Standardization and Transformation 33
3.2.1.1 Z-score Standardization 34
3.2.1.2 Min-Max Standardization 34
3.2.1.3 Decimal Scaling Standardization 35
3.2.2 Principal Component Analysis 35
3.3 Singular Value Decomposition 38
3.3.1 Components To be Retained 40
3.4 K-means Cluster Analysis 41
3.5 The Proposed Hybrid K-means Method 42
3.6 How To Obtain Initial Cluster Centers 48
3.7 Chapter Summary 49
4 METHOD IMPLEMENTATION 50
4.1 Introduction 50
4.2 Data Pre-processing 50
4.2.2 K-means with Preprocessed Data 59
4.2.3 Discussion 63
4.3 Chapter Summary 64
5 EVALUATION OF THE HYBRID K-MEANS METHOD 65
5.1 Introduction 65
5.2 Covariance Matrix Structure 65
5.3 Simulation Experiment with Uncorrelated Dataset 67 5.4 Simulation Experiment with Correlated Dataset 70
5.4.1 Simulation from generated dataset where correlation is
controlled 70
5.4.2 Simulation from generated data set with fixed
correlation 78
5.4.3 Discussion 81
5.5 Separable Cases 81
5.5.1 Discussion 85
5.6 Comparative Analysis for the Basic K-means and Hybrid K-means 85
5.6.1 Discussion 91
5.7 Z-test for < 91
5.7.1 Discussion 96
5.8 An Example to Illustrate the Hybrid K-means Method 97
5.9 Chapter Summary 104
6 CASE STUDY 105
6.1 Introduction 105
6.2 Pattern of Infection Across Countries 106
6.3 Cluster Formations 106
6.4 Discussion 111
6.5 Recommendation 112
7 CONCLUSION 113
7.1 Introduction 113
x
7.3 Achievement 114
7.4 Further Research 115
REFERENCES 117
LIST OF TABLES
TABLE NO. TITLE PAGE
4.1 The Rafindadi Clinic data 51
4.2 The Standardized z-score for Rafindadi Clinic data 54 4.3 The Standardized decimal scaling for Rafindadi Clinic data 56 4.4 The Standardized min-max for Rafindadi Clinic data 58
4.5 The PCs variances and cumulative percentage 60
4.6 The reduced principal components of the Rafindadi Clinic data 61
4.7 Projected data 62
4.8 Summary of cluster formation results 63
5.1 The total sum of squares error and time taken for uncorrelated datasets 70 5.2 The total sum of squares errors and time taken for computed correlated
dataset 78
5.3 The total sum of squares errors and time taken for correlated datasets 80 5.4 Error sum of squares for the separable non separable, half Separable and
Separable 85
xii 5.8 The number of time and proportions of < for = 7 from 500 simulations (The numbers in brackets are the proportions and * symbols indicates the proportion is significantly large than 50% at =0.05) 94
5.9 The original sample data 97
5.10 The Standardized z-score for the sample data 99
5.11 The The PCs variances and cumulative percentage of the standardized
sample data 99
5.12 The principal components of the standardized sample data 100 5.13 The reduced principal components of the standardized sample data 100
5.14 Reduced projected data 101
5.15 Cases number and clusters 102
LIST OF FIGURES
FIGURE NO. TITLE PAGE
2.1 Clustering Process 10
2.2 Research Framework 31
4.1 Basic K-means centroids 52
4.2 Z-score basic K-means centroids 55
4.3 Decimal scaling basic K-means centroids 57
4.4 Min-Max basic K-means centroids 59
4.5 PCA/SVD basic K-means centroids 62
5.1 Basic K-means centroids for uncorrelated data (5, 500) 68 5.2 Hybrid K-means centroids for uncorrelated data (5, 500) 68 5.3 Basic K-means centroids for uncorrelated data (20, 500) 69 5.4 Hybrid K-means centroids for uncorrelated data (20, 500) 69
5.5 Basic K-means centroids V1(5, 50) 72
5.6 Hybrid K-means centroids V1(5, 50) 72
5.7 Basic K-means centroids V2(5, 50) 73
5.8 Hybrid K-means centroids V2(5, 50) 74
5.9 Basic K-means centroids V3(5, 50) 75
5.10 Hybrid K-means centroids V3(5, 50) 76
5.11 Basic K-means centroids V4(5, 50) 77
5.12 Hybrid K-means centroids V4(5, 50) 77
xiv 5.19 Basic K-means centroids in the half separable case 83 5.20 Hybrid K-means centroids in the half separable case 83
5.21 Basic K-means centroids in the separable case 84
5.22 Hybrid K-means centroids in the separable case 84
5.23 Clustering results for (5, 500) and K=2 87
5.24 Clustering results for (10, 500) and k=3 88
5.25 Clustering results for (20, 500) and K=5 89
5.26 Clustering results for (50, 500) and K=7 90
5.27 Basic K-means method 103
5.28 Hybrid K-means method 103
6.1 Classification of KGHK data into two clusters 110
LIST OF ABBREVIATIONS
CPU - Central Processsor Unit
DVSCAN - Density-Based Clustering Algorithm
FE - Feature Extraction
FMCK - Federal Medical Centre, Katsina
FR - Feature Reduction
GA - Genetic Algorithm
GAIK - Genetic Algorithm Initialize K-means
GCA - Genetic-Baseb Clustering Algorithm
IGK - Improved Genetic K-means
KGHK - Katsina General Hospital, Katsina
KIGA - K-means Initializes the Genetic Algorithm
LOF - Local Outlier Factor
PC - Principal Component
PCA - Principal Component Analysis
PD - Partial Distance
SCP - Spectral Constaraint Prototype
SSE - Sum of Squares Error
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LIST OF APPENDICES
APPENDIX TITLE PAGE
A Publications 125
B Data Collected 127
CHAPTER 1
INTRODUCTION
1.1 Background of the Study
The K-means clustering algorithm is one of the most popular methods for clustering multivariate observations (Tsai and Chiu, 2008). It is a system ordinarily used to directly segment sets of data into k groups. K-means algorithm generates a fast and efficient solution. The basic K-means algorithm works with the objective to minimize the mean square distance from each data point to its nearest center.
There are two important issues in creating a K-means clustering algorithm: the optimal number of clusters and the center of the cluster. In many cases, the number of clusters is given, thus the important issue is where to put the cluster center so that scattered points can be grouped appropriately. Center of the cluster can be obtained by first assigning any random point and then optimizing the mean distance to the center. The process is repeated until all the mean square distances are optimized.
2 calculations increases exponentially with the increase of the dimensionality of the data. An ad-hoc solution to these problems is by choosing a set of different initial partition and the initial partition that gives the smallest sum of squares error is taken as the solution but this ad-hoc solution does not guarantee the solution will give the smallest sum of square error (SSE) because it is just a mere guessing approach.
When a random initialization of centroids is used, different runs of K-means typically produce different total SSEs, therefore choosing the proper initial centroids is the key step of the basic means procedure (Zhu et al. 2009). The result of the K-means algorithm is highly dependent upon its initial selection of cluster centers and before clustering it must be previously known and fixed (Tsai and Chiu, 2008). Fahim et al. (2009) proposed a method to select a good initial solution by partitioning data set into blocks and applying K-means to each block. But here the time complexity is slightly more.
Tajunisha and Saravanan (2010) proposed a method to improve the performance of the K-means algorithm, using principal component analysis (PCA) for dimension reduction and to find the initial centroid for K-means. The method partitioned the data set into K sets and the median of each set were used as initial cluster centers and then assign each data point to its nearest cluster centroid. Heuristic approach was also used to reduce the number of distance calculation in the standard K-means algorithm to assign the data point to the cluster.
To obtain an optimum solution for K-means clustering, the data need to be pre-processed before the K-means clustering analysis (Chandrasekhar et al. 2011). This pre-processing process consists of data standardization method to rescale the dataset and principal component analysis method foroutliers’detection. Outliers are the data that are numerically distant from the rest of the data. If they are not properly detected and handled, the clustering result will be affected in a great manner (Sairam et al. 2011).
An approach to handle outlier is data standardization, it rescale the data set to fall within a specified range of values so that any attribute with larger value will not dominate the attribute with a smaller value. However, for a very high dimensional data set, PCA can be used to initially reduce this dimension (Chris and Xiaofeng, 2006). They further proved that principal components are the continuous solutions to the discrete cluster membership indicators for K-means clustering and showed that unsupervised dimension reduction is closely related to unsupervised learning. On dimension reduction, the result provides new insights to the observed effectiveness of PCA-based data reductions, beyond the conventional noise-reduction explanation.
As K-means is highly dependent on its initial center position (Rana et al. 2010), an alternative way of center initialization method for K-means cluster analysis is also required to make the algorithm more effective and efficient. To overcome the above drawback the current research focused on developing a K-means clustering technique by data preprocessing and the initialization of the center points for high dimensional datasets.
1.2 Problem Statement
4 may not be accurate due to noise and outliers associated with the initial dataset. This brings about an increase in computational complexity and also resulting in some attributes with larger domain dominating those attributes with lower domain. Outliers are the data that are numerically distant from the rest of the data and if they are not detected and handled, they tend to affect the clustering result.
In the creation of a K-means clustering algorithm two main issues are prominent, these are: the optimal number of clusters and the cluster center points. In most cases, the number of clusters is given, thus leaving the challenge where to put the cluster centers so that scattered points can be grouped properly and to avoid its convergence to a local minimum of the objective function. Furthermore, the random initialization results in different total SSEs value from several runs of the K-means. This makes the result from the algorithm of the K-means to depend greatly on the initial selection of the cluster centers which must be known and fixed beforehand. Therefore, the choice of proper initial centroids is pertinent to the basic K-means procedure.
As fallout from the above, a new technique of centers initialization for K-means clustering is required to make the algorithm more effective and efficient. Hence this research focus on an alternative way of handling the K-means clustering technique by data pre-processing and the initialization of the center points for high dimensional datasets.
1.3 Objectives of the Study
The second part consists of the followings:
i. Centers initialization using singular value decomposition (SVD) to avoid random initialization and convergence to a local minimum, this will make the algorithm more effective and efficient.
ii. Simulation experiment and comparison of the optimality for the basic and the proposed techniques.
The third part is the application of the technique to Nigerian data of infectious diseases to demonstrate the use of this technique.
1.4 Scope of the Study
This research covers the following three aspects, computational (data pre-processing), initialization of centre points, and practical aspects. Furthermore, the number of clusters required is always pre-determined throughout the thesis.
i. Computational aspect (Data pre-processing)
In computational aspects, we used a real data of infectious diseases consisting of seven variables and a sample size of 20 to come up with a good method of data pre-processing for K-means cluster analysis and a simulation experiments to validate and test the proposed hybrid K-means technique.
ii. Centre point initialization aspect
6 iii. Practical aspect
Application in real data of infectious diseases from hospital was used to show the performance and advantage of the hybrid K-means method developed in this thesis.
1.5 Contribution of the Study
This thesis offers a contribution in two aspects: 1. The main contributions are
i. New method in choosing the initial centroids for K-mean cluster analysis.
ii. A technique of K-means clustering algorithm for high dimension dataset. Simulation experiments proved that the proposed technique is optimum and converges faster than the basic method.
iii. The computational complexity of the proposed K-means clustering technique is far lower than the basic K-means algorithm. This is another advantage of the proposed method especially when the data sets are of higher dimensions.
2. This thesis used infectious diseases data sets. Therefore we hope that the finding results will be useful and assist the government to embark on health policy formulations that will address the problems of the intensity of diseases in the states and to plan adequately for the provision of welfare services such as good pipe borne water, good drainage system and environmental protection.
1.6 Thesis Organization
clusters and the center of the cluster. The drawback of the basic K-means algorithm is also stated, the research objectives are defined, and the scope of this research is also presented. However, the chapter presents the significance of this study and ends with the contribution of the study.
Chapter 2 presents a comprehensive literature review. The target is to review current literatures to identify what has previously been attained and recognize the gap in our research. The existing theories of the K-means clustering technique, similarity measures, principal component analysis, data standardization methods and the evolution of ideas in the initialization of the K-means cluster centers are presented, this ends with a research framework.
Chapter 3 focuses on an in depth explanation of the methodologies. It concentrates on how this research is carried out in order to arrive at the findings and conclusions. This chapter describes the data pre-processing methods and how to implement the proposed K-means technique.
Chapter 4 concentrated on the data pre-processing methods, implementation of the proposed hybrid K-means clustering method and validation of the method using simulation experiment with correlated and uncorrelated in Chapter 5. After that, Chapter 6 demonstrates the use of this method to Nigerian data of infectious diseases. This thesis closes with the conclusion and recommendation for future research in Chapter 7.
1.7 Chapter Summary
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